Topological Derivatives and other Embeddings for Ocean Floor Tsunami data

1. Topological Derivatives and other Embeddings for Ocean Floor Tsunami data Prabhakar G. Vaidya, Nithin Nagaraj, Sajini Anand P S Mathematical Modelling Unit National Institute of Advanced Studies IISc. Campus, Bangalore-12 (research partly supported by ISRO and DST) International Conference on Non-linear Waves and Tsunami March 6-10, Kolkatta

2. Role of Embedding in non-linear signal processing • Topological embedding • Modelling • Detection • Noise Removal • Prediction

3. Object of Dimension D P RD … Injection no inverse Q Topologically conjugate Object RN … What is Embedding? Q’

4. Takens Embedding Scalar variables measured at periodic interval Linear Systems: For D dimensional dynamics, almost always embedding dimension N = D, where N is the dimension of the data vector Non-linear systems: Takens theorem(1981): N>2D

5. Topological Conjugacy • Embedding ensures that there is a continuous smooth map from embedded variables to the original variables. • All the bifurcations and other properties are mimicked in the embedded picture • Derivative embedding • Close to physical Intuition • Prone to serious errors due to noise

6. Topological Derivatives • Topological derivatives are the same as exact derivatives for • i) No Noise • ii) Low dimensional dynamics • iii) Very high sampling rate • For noisy data: • Noise is reduced • For low sampling rate, it is diffeomorphic with actual derivatives. • Yields low dimensional or the best low dimensional dynamics. • Ref: P. G. Vaidya, Monitoring and speeding up chaotic synchronization, Chaos, Solitons and Fractals, Volume 17, Number 2, July 2003, pp. 433-439(7)

7. De-tiding of Tide-gauge data High-Pass filter output for de-tiding (cut-off freq. = 0.314 Hz) Tide gauge data Power spectrum of data Low-Pass filter output for de-spiking (cut-off freq. = 0.628 Hz) Period=24 hours (Earth rotation) Period=12 hours (Moon revolution) • De-tiding involves calculating the period contributed by the Earth’s rotation and the Moon’s revolution which results in tides and removing them from signal. We have done this in the Fourier domain using a High-Pass filter. *Source: West Coast/Alaska Tsunami Warning Center: http://wcatwc.arh.noaa.gov/IndianOSite/IndianO12-26-04.htm

8. Ocean Floor Data Analysis Bottom Pressure Recorded Data* (sampling rate: 56.25 secs) High pass filter output (cut-off frequency is lower 6% of the spectrum ) *Source: Data obtained from National geophysical data center (http://www.ngdc.noaa.gov/seg/hazard/tsu.shtml)

9. Topological Derivative Embedding Evolution of the signal *Source: Data obtained from National geophysical data center (http://www.ngdc.noaa.gov/seg/hazard/tsu.shtml)

10. Time evolution of the position and velocity vectors

11. Disscussion and conclusions • Embedding started as a way to get Pictures from data (Farmer ,Packard 1980) • We have shown how from a single variable using topological derivatives and other methods we can get state space plots • These have a potential for helping detecting Tsunami’s in deepwater even in the presence of large noise